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Well I am glad you asked. I am an EUNE currently Silver III summoner. I have been playing Ezreal since Season 3 and he quickly became one of my main champions. At the beginning I played as ADC Ezreal. Then I tried the legendary Blue Build. Which was really fun but unfortunately now outdated and not viable. In Season 4 I heard that Ezreal was originally a mage and became an ADC in the pass of time and patches. So in a guide me and my fried tried out troll builds of AP EZ and AP Shaco (gonna make a guide on that soon). Turns out both of them are far from just viable. Since then AP Ez is the only way I play Ez. This playstyle has great advantages ADC Ezreal does not have.

Pros!:

+Really, really freaking powerful ult!
+Great Range with abilities
+Extremely mobile
+Great repositioning, engaging and disengaging with E
+Lane Clearing

Cons:

-No Escape when E is on cooldown
-All abilities are skillshots
-Really farm reliant
-Some people think he's a girl O.O

Don't get me wrong here. I love all Ez builds, AP, AD, even blue builds. But AP has these advantages:

1. Your main harass skillshot can't be blocked by minions. This means great harass and bursting down or forcing champions out of lane.

2. Ulti Spamming! Yes with AP ez you can easily spam your ult using Arch Angels staff. You know how? When you get your Arch Angels Staff you have infinite mana. You can't go OOM. So you start spamming your Q on minions. This helps in getting easy CS and lowering your ultimate's cooldown to ridiculous rate.

3. Ultimate is TOO FREAKING POWERFUL! Your ultimate scales more with AP and becomes so powerful from level 11 and up! There was a game a while ago in which I one shotted with my ultimate a full HP Heimerdinger (my ult hit him first).

This is your bread and butter as Ezreal. It gives you mainly AP but the reason you buy this is the passive. After a skillshot the next basic attack hits really hard. This is great for farming and fights